Consider the case where the computerization described in paragraph five has been implemented. So, the average service time for the counters engaged in accountable transactions has been reduced from 2.5 minutes to 1.5 minutes. Also, the need for customers to buy stamps for accountable transactions has also been eliminated.

a) Assume that there is a common queue for accountable transactions. In other words, assume a separate queue for each of the following: 1) Counter for savings bank services (SB); 2) Both counters for accountable services (A1&A2 combined, so that there is only 1 queue for accountable services); and 3) Counter for stamps and stationary services (SS).

1. Calculate a customer's expected waiting time in each of the three queues

2. Calculate the average number of customers in each of the three queues

3. Calculate a customer's total expected time in the system for each queue

4. Calculate the average time a customer spends in the post office, averaged across all customers.

b) Now assume that there is only one counter for accountable transactions, so that there is a separate queue for each of the following: 1) Counter for savings bank services (SB); 2) One counter for accountable services (A); and 3) Counter for stamps and stationary services (SS).

1. Calculate a customer's expected waiting time in each of the three queues

2. Calculate the average number of customers in each of the three queues

3. Calculate a customer's total expected time in the system for each queue

4. Calculate the average time a customer spends in the post office, averaged across all customers.

Swami and Friends at the Marguar Post Orrice The Malgudi' post office received an average of 60 customers per hour seeking a variety of products and services, sach as stamps and stationery, savings bank (SB) services and "accountable" services, i.e, registration, speed poot, money onders and telegrams, Of the 60, an average of 10 came primarily for SB transactions and 25 primarily for accountable services. The remaining 25 came exclusively for stamps and stationery. Around 15 customers who came for accountable services usually stood in line thrice: once to find out what they needed to do and the value of stampo needed; then to buy the stamps; and finally, to do the accountable transaction. About half of the custumers coming to the post office for accountable and SB transactions invariably required stamps and stationery. The post office had one $8 counter, two counters for accountable transactions and one counter for stamps and stationery. The average service time at the SB counter, accountable service counters and the stamps and stationery coumter was three minutes, two and a half minutes and one minute, respectively. Swami, the postmaster, observed that the stamps and stationery line was usually the longest. However, he received the maximum complaints from castomers who came for sccountable transactions. He was wondering what he could do to improve customer service, especially in terms of reducing waiting times, when he received a letter from Narayanan, the Superintendent of Post Offices. The letter indicated that Malgudi had been selected as one of the post offices for computerisation. Computerisation was expected to enable the counters engaged in accountable transactions to reduce the average service time from two and a half minutes to one and a half minutes. It would also eliminate the need for customers to buy stamps for accountable transactions. The letter also indicated the possibility of introducing an improved sottware that would enable S8 transactions to be processed as well. However, since the counter derks would have to adapt to a wider range of activities, the average service time would now be two minutes. But the counters would now be truly multi-purpose. Swami thought that this would indeed be a boon since there was potential to improve the performance of both the SB and accountable service counters significantly. However, he soon tealised that there was a eatch in the proposal: the post office wothld have to reorganise itself to release one counter derk for redeployment elsewhere. He felt dejected and wondered whether there would be any improvement at all. (Table 1 could be used for the analysis) Table 1: Average Number in Waiting Line (Nq), With 95th and 99th Percentiles* \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline Utilisation Factor & 0.10 & 0.20 & 0.30 & 0.40 & 0.50 & 0.60 & 0.70 & 0.75 & 0.80 & 0.85 & 0.90 & 0.95 & 0.98 \\ \hline \multicolumn{14}{|l|}{ Servers } \\ \hline \multirow[t]{2}{*}{15} & 0.000 & 0.000 & 0.000 & 0.001 & 0.011 & 0.072 & 0.329 & 0.654 & 1.277 & 2.533 & 5.424 & 14.952 & 44.656 \\ \hline & 00 & 00 & 00 & 00 & 00 & 03 & 27 & 510 & 815 & 1323 & 2338 & 5385 & 143223 \\ \hline \multirow[t]{2}{*}{16} & 0.000 & 0.000 & 0.000 & 0.001 & 0.009 & 0.063 & 0.303 & 0.614 & 1.220 & 2.455 & 5.322 & 14.824 & 44.510 \\ \hline & 00 & 00 & 00 & 00 & 00 & 02 & 27 & 410 & 815 & 1323 & 23.38 & 5384 & 143223 \\ \hline \multirow[t]{2}{*}{17} & 0.000 & 0.000 & 0.000 & 0.000 & 0.007 & 0.055 & 0.278 & 0.577 & 1.166 & 2.381 & 5.225 & 14.700 & 44,369 \\ \hline & 00 & 00 & 00 & 00 & 00 & 02 & 26 & 410 & 715 & 1323 & 2338 & 5384 & 143223 \\ \hline \multirow[t]{2}{*}{18} & 0.000 & 0.000 & 0.000 & 0.000 & 0.006 & 0.048 & 0.256 & 0.542 & 1.116 & 2.312 & 5.132 & 14.58 & 44.232 \\ \hline & 00 & 00 & 00 & 00 & 00 & 02 & 26 & 410 & 714 & 1222 & 2338 & \begin{tabular}{ll} 53 & 84 \end{tabular} & 143222 \\ \hline \multirow[t]{2}{*}{19} & 0.000 & 0.000 & 0.000 & 0.000 & 0.005 & 0.041 & 0.236 & 0.511 & 1.069 & 2.245 & 5.043 & 14.465 & 44.099 \\ \hline & 00 & 00 & 00 & 00 & 00 & 01 & 16 & 49 & 714 & 1222 & 2238 & 5384 & 143222 \\ \hline \multirow[t]{2}{*}{20} & 0.000 & 0.000 & 0.000 & 0.000 & 0.004 & 0.036 & 0.218 & 0.481 & 1.024 & 2.182 & 4.957 & 14.353 & 43.970 \\ \hline & 00 & 00 & 00 & 00 & 00 & 01 & 16 & 49 & 714 & 1222 & 2238 & 5284 & 142222 \\ \hline \multirow[t]{2}{*}{25} & 0,000 & 0.000 & 0.000 & 0.000 & 0.001 & 0.019 & 0.148 & 0.362 & 0.836 & 1.905 & 4.571 & 13.839 & 43.371 \\ \hline & 00 & 00 & 00 & 00 & 00 & 00 & 05 & 38 & 613 & 1121 & 2237 & 5283 & 142221 \\ \hline \multirow[t]{2}{*}{30} & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.010 & 0.102 & 0.276 & 0.691 & 1.680 & 4.243 & 13,386 & 42.834 \\ \hline & 00 & 00 & 00 & 00 & 00 & 00 & 04 & 27 & 512 & 1020 & 2136 & 5182 & 141221 \\ \hline \multirow[t]{2}{*}{35} & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.005 & 0.072 & 0.213 & 0.577 & 1.491 & 3.957 & 12.979 & 42.344 \\ \hline & 00 & 00 & 00 & 00 & 00 & 00 & 03 & 16 & 411 & 1020 & 2035 & 5082 & 141220 \\ \hline \multirow[t]{2}{*}{40} & 0,000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.003 & 0.051 & 0.166 & 0.485 & 1.331 & 3.704 & 12.609 & 41.892 \\ \hline & 00 & 00 & 00 & 00 & 00 & 00 & 02 & 05 & 311 & 919 & 2035 & 5081 & 140220 \\ \hline \multirow[t]{2}{*}{50} & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.001 & 0.026 & 0.102 & 0.348 & 1.075 & 3.275 & 11.953 & 41.075 \\ \hline & 00 & 00 & 00 & 00 & 00 & 00 & 00 & 04 & 29 & 818 & 1824 & 4980 & 139219 \\ \hline \end{tabular}